scholarly journals Polyharmonic Problems with Simply Supported Polynomial Data in the Unit Ball in $ R^n$

2018 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Boundary Value Problem ◽  
Unit Ball ◽  
Poisson Equation ◽  
Boundary Function ◽  
Exact Algorithms ◽  
Dirichlet Problems ◽  
Polyharmonic Equation ◽  
Dirichlet Conditions ◽  
Simply Supported ◽  
Laplace Equations

We studied simply supported polynomial data of boundary value problem of Polyharmonic equation. The problem is reformulated as a systems of Laplace-Poisson equation with Polynomial Dirichlet problems. We utilize an exact algorithms for solving Laplace equations with Dirichlet conditions with polynomial data. The algorithm requires differentiation of the boundary function, but no integration.

scholarly journals Tri-Dirichlet-Type Problems with Polynomial Data in a Unit Sphere In $ R^3$

2017 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Boundary Value Problem ◽  
Poisson Equation ◽  
Unit Sphere ◽  
Dirichlet Boundary ◽  
Boundary Function ◽  
Exact Algorithms ◽  
Polynomial Functions ◽  
Dirichlet Boundary Value Problem ◽  
Dirichlet Problems ◽  
Laplace Equations

We studied Tri-Dirichlet boundary value problem of TriLaplace equation. The problem is reformulated as a systems of Laplace-Poisson equation with Dirichlet problems. We utilize an exact algorithms for solving Laplace equations with Dirichlet conditions with polynomial functions data. The algorithm requires differentiation of the boundary function, but no integration.

scholarly journals Biharmonic Problems with Polynomial Data in a Unit Sphere In $ R^3$

2018 ◽  
Author(s):  
Agah D. Garnadi ◽  
Ikhsan Maulidi
Keyword(s):  
Boundary Value Problem ◽  
Unit Sphere ◽  
Biharmonic Equation ◽  
Boundary Function ◽  
Exact Algorithms ◽  
Polynomial Functions ◽  
Dirichlet Problems ◽  
Simply Supported ◽  
Laplace Equations ◽  
Biharmonic Problems

We studied simply supported boundary value problem of Biharmonic equation. The problem is reformulated as a systems of Laplace-Poisson equation with Dirichlet problems. We utilize an exact algorithms for solving Laplace equations with Dirichlet conditions with polynomial functions data. The algorithm requires only differentiation of the boundary function, but no integration

scholarly journals Simply Supported Biharmonic Equations with Polynomial Data in The Unit Ball of $ R^n$

2018 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Unit Ball ◽  
Dirichlet Boundary ◽  
Exact Algorithms ◽  
Dirichlet Boundary Conditions ◽  
Polynomial Functions ◽  
Dirichlet Problems ◽  
Biharmonic Equations ◽  
Simply Supported ◽  
Poisson Equations ◽  
Laplace Equations

We studied simply supported boundary value problem of Biharmonic equation in the unit ball of $R^n, n \geq 3,$ with polynomial data. The problem is restated as a pair of Laplace and Poisson equations with polynomial Dirichlet problems. We utilize an exact algorithms for solving Laplace equations with Dirichlet boundary conditions with polynomial functions data. The algorithm requires only differentiation of the boundary data, but no integration

scholarly journals Biharmonic Equation on Annulus in a Unit Sphere with Polynomial Boundary Condition

2017 ◽  
Vol 13 (1) ◽  
pp. 51
Author(s):  
Ikhsan Maulidi ◽  
Agah D Garnadi
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Unit Sphere ◽  
Biharmonic Equation ◽  
Boundary Value ◽  
Boundary Function ◽  
Exact Algorithms ◽  
Polynomial Functions ◽  
Dirichlet Conditions ◽  
Laplace Equations

We studied Biharmonic boundary value problem on annulus with polynomial data. We utilize an exact algorithms for solving Laplace equations with Dirichlet conditions with polynomial functions data. The algorithm requires differentiation of the boundary function, but no integration.

scholarly journals Biharmonic Equation on Annulus in a Unit Sphere with Polynomial Boundary Condition

2017 ◽  
Vol 13 (1) ◽  
pp. 51
Author(s):  
Ikhsan Maulidi ◽  
Agah D Garnadi
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Unit Sphere ◽  
Biharmonic Equation ◽  
Boundary Value ◽  
Boundary Function ◽  
Exact Algorithms ◽  
Polynomial Functions ◽  
Dirichlet Conditions ◽  
Laplace Equations

We studied Biharmonic boundary value problem on annulus with polynomial data. We utilize an exact algorithms for solving Laplace equations with Dirichlet conditions with polynomial functions data. The algorithm requires differentiation of the boundary function, but no integration.

scholarly journals Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1907
Author(s):  
Valery Karachik
Keyword(s):  
Boundary Value Problem ◽  
Unit Ball ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Dirichlet Boundary Value Problem ◽  
Polyharmonic Equation ◽  
Harmonic Components

In the previous author’s works, a representation of the solution of the Dirichlet boundary value problem for the biharmonic equation in terms of Green’s function is found, and then it is shown that this representation for a ball can be written in the form of the well-known Almansi formula with explicitly defined harmonic components. In this paper, this idea is extended to the Dirichlet boundary value problem for the polyharmonic equation, but without invoking the Green’s function. It turned out to find an explicit representation of the harmonic components of the m-harmonic function, which is a solution to the Dirichlet boundary value problem, in terms of m solutions to the Dirichlet boundary value problems for the Laplace equation in the unit ball. Then, using this representation, an explicit formula for the harmonic components of the solution to the Neumann boundary value problem for the polyharmonic equation in the unit ball is obtained. Examples are given that illustrate all stages of constructing solutions to the problems under consideration.

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

scholarly journals The Gradient of a Solution of the Poisson Equation in the Unit Ball and Related Operators

2017 ◽  
Vol 60 (3) ◽  
pp. 536-545 ◽  
Author(s):  
David Kalaj ◽  
Djordjije Vujadinovic
Keyword(s):  
Integral Operator ◽  
Unit Ball ◽  
Poisson Equation ◽  
Boundary Data ◽  
The Poisson Equation

AbstractIn this paper we determine the L1 ⟶ L1 and L∞ ⟶ L∞ norms of an integral operator N related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.

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